Chapter 6. Electromagnetic Oscillations and Alternating Current

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1 hapter 6 Electromagnetic Oscillations an Alternating urrent

2 hapter 6: Electromagnetic Oscillations an Alternating urrent (hapter 31, 3 in textbook) 6.1. Oscillations 6.. The Electrical Mechanical Analogy 6.3. Dampe Oscillations in an ircuit 6.4. Alternating urrent 6.5. Three Simple ircuits 6.6. The Series ircuit 6.7. Power in Alternating urrent ircuits 6.8. Transformers

3 Overview n the previous chapters, we have stuie the basic physics of electric an magnetic fiels an how energy can be store in capacitors an inuctors. n this chapter, we will stuy the associate applie physics, in which the energy store in one location can be transferre to another location so that can be put to use, e.g., energy prouce at a power plant can be transferre to your home to run a computer. The moern civilization woul be impossible without this applie physics.

Boiler Turbine Generator Electricity cables Fuel Furnace ooling tower Step-up transformer Step-own transformer Pylons Appliances Homes n most parts of the worl, electrical energy is transferre not as

4 Boiler Turbine Generator Electricity cables Fuel Furnace ooling tower Step-up transformer Step-own transformer Pylons Appliances Homes n most parts of the worl, electrical energy is transferre not as a irect current but as a sinusoially oscillating current (alternating current, or ac). The challenge to us is to esign ac systems that transfer energy efficiently an to buil appliances that make use of that energy.

5 6.1. Oscillations: Oscillations, Qualitatively: n an circuits the charge, current, an potential ifference grow an ecay exponentially. On the contrary, in an circuit, the charge, current, an potential ifference vary sinusoially with perio T an angular frequency. The resulting oscillations of the capacitor s electric fiel an the inuctor s magnetic fiel are sai to be electromagnetic oscillations.

6 8 stages in a single cycle of oscillation of a resistance less circuit

7 The energy store in the electric fiel of the capacitor at any time is q U E where q is the charge on the capacitor at that time. The energy store in the magnetic fiel of the inuctor at any time is U B i where i is the current through the inuctor at that time. As the circuit oscillates, energy shifts back an forth from one type of store energy to the other, but the total amount is conserve.

the time-varying potential ifference v across it: v i n actual circuits, the oscillations o not

8 The time-varying potential ifference (or voltage) v that exists across the capacitor is: v 1 q To measure i, we can connect a small resistance in series with the capacitor an inuctor then measure the time-varying potential ifference v across it: v i n actual circuits, the oscillations o not continue forever because some resistance will issipate electrical an magnetic energy as thermal energy

9 6.. The Electrical Mechanical Analogy: One can make an analogy between the oscillating system an an oscillating block spring system. Here we have the following analogies: Thus, The angular frequency: k m (block spring system) 1 ( circuit)

10 The Block-Spring Oscillator: 1 1 U U U mv kx b s U 1 1 v x mv kx mv kx 0 t t t t x m kx 0 x X cos( t ) (isplacement) t The Oscillator: U U t U t q B U E i t 1 q i q q i i t 0 ( oscillations) i Prove the angular frequency of q q t : phase constant q t 0 ; k m q Qcos( t ) (charge) Q sin( t ) (current)

11 The amplitue : Q i sin( t ) Angular Frequencies: The equation to calculate charge q is inee the solution of the oscillations equation with 1/, we can test: q Qcos( t ) We have t t q 1 q 0 ( oscillations) Qcos( t ) 1 1 Qcos( t ) 0

Electrical an Magnetic Energy Oscillations: The electrical energy store in the circuit at time t is, q Q U cos E ( t ) The magnetic energy is: 1 1 U i Q sin B But Therefore: 1 U B ( circuit)

12 Electrical an Magnetic Energy Oscillations: The electrical energy store in the circuit at time t is, q Q U cos E ( t ) The magnetic energy is: 1 1 U i Q sin B But Therefore: 1 U B ( circuit) Q sin ( t ) ( t ) Note that: The maximum values of U E an U B are both Q /. At any instant the sum of U E an U B is equal to Q /, a constant. When U E is maximum, U B is zero, an conversely.

ecreases with time as energy is transferre to thermal energy in the resistance.

13 6.3. Dampe Oscillations in an ircuit: A circuit containing resistance, inuctance, an capacitance is calle an circuit With, the total energy U of the circuit (the sum of U E an U B ) is not constant; instea, it ecreases with time as energy is transferre to thermal energy in the resistance. The oscillations of charge, current, an potential ifference continuously ecrease in amplitue, an the oscillations are sai to be ampe.

14 Analysis: U U B U The total energy ecreases as U transferre to thermal energy substituting q/t for i an q/t for i/t, we have: E i U i q q i i t t t t q q Qe q 1 q 0 ( circuit) t t / q cos( ' t ) Where ' ( / ) An 1/ U E q Qe t / cos( ' t ) Q e t / cos ( ' t )

6.4. Alternating urrent (A): The oscillations in an circuit will not amp out if an external emf evice supplies enough energy to make up for the energy issipate as thermal energy The basic

15 6.4. Alternating urrent (A): The oscillations in an circuit will not amp out if an external emf evice supplies enough energy to make up for the energy issipate as thermal energy The basic mechanism of an A generator is a conucting loop rotate in an external magnetic fiel The inuce emf: m sin t i sin( t ) is calle the riving angular frequency, an is the amplitue of the riven current.

16 Force Oscillations: Unampe circuits an ampe circuits oscillate at: 1/ that is natural angular frequency When the circuit is connecte to an external alternating emf, charge, potential ifference an current oscillate with riving angular frequency, these oscillations are the so-calle riven oscillations or force oscillations The amplitue of the current in the circuit is maximum when =, a conition known as resonance A generator, represente by a sine wave in a circle, prouces an alternating emf that establishes an alternating current; the irections of the emf an currents are inicate here at only instant.

17 6.5. Three Simple ircuits: A esistive oa: v 0 v i m v sin t sin t sin t Note: for a purely resistive loa the phase constant = 0. v (t) an i (t) are in phase ( = 0 o ), which means their corresponing maxima (an minima) occur at the same times. The time-varying quantities v an i can also be represente geometrically by phasors (vectors that rotate aroun an origin) with the following properties: angular spee (rotate counterclockwise with ). length (representing the amplitue of an ) projection (representing v an i on the vertical axis); rotation angle (to be equal to the phase t at time t)

19 6.5.. A apacitive oa: v q i v q t X sin t sin t cos 1 X is calle the capacitive reactance of a capacitor. The S unit of X is the ohm (), just as for resistance. cos t sin( t 90 i sin( t 0 t (capacitive ) ), i 0-90 X reactance) sin( t 90 0 ) X (capacitor)

21 An nuctive oa: Potential ifference across the inuctance: From: i i i t (1) an () ntegrating: X i sin( t i t v v sin t X, the inuctive reactance, epens on the riving angular frequency. The unit of the inuctive time constant (=/) inicates that the S unit of X is the ohm. i t sin tt sin cos t X (inuctivereactance) 90 0 ); i () (1) t sin( t ) X (inuctor)

24 6.6. The Series ircuit: () The emf phasor is equal to the vector sum of the three voltage phasors of (b).here, voltage phasors an have been ae vectorially to yiel their net phasor ( - ).

25 m t sin sin( ) t i m X X ( ) m X X (impeance efine) ) ( X X Z (current amp litue) ) 1/ ( m X X tan (phaseconstant) tan X X

27 esonance: ( m 1/ ) (current amp litue) For a given resistance, that amplitue is a maximum when the quantity ( -1/ ) in the enominator is zero. 1 1 (maximum ) The maximum value of occurs when the riving angular frequency matches the natural angular frequency that is, at resonance. 1 (resonance)

28 esonance curves for a riven circuit with 3 values of. The horizontal arrow on each curve measures the curve s half-with, which is the with at the half-maximum level an is a measure of the sharpness of the resonance. X >X X <X

29 The Parallel ircuit: m t sin - t i m sin ( ) X X Z Z tan 1 1 tan

30 6.7. Power in A ircuits: The instantaneous rate at which energy is issipate in the resistor: P i sin( t ) sin ( t ) The average rate (average power): P avg rms P avg rms oot-mean-square: rm s P P avg avg rm s Z rms Z rms rms rms rms rm s an ( X X ) cos rms rms rms Z m Where: cos m Z Z

31 6.8. Transformers: n electrical power istribution systems, for reasons of safety an efficient equipment esign to eal with relatively low voltages at both the generating en (the electrical power plant) an the receiving en (the home or factory). For ex: Noboy wants an electric toaster or a chil s electric train to operate at 10 k. On the other han, in the transmission of electrical energy from the generating plant to the consumer, we want the lowest practical current (hence the largest practical voltage) to minimize losses (often calle ohmic losses) in the transmission line.

32 Example: onsier a 500 k line use to transmit electrical energy from Hòa Bình to HM city, 1500 km away. Suppose that the current is 500 A an the power factor is close to unity. So, energy is supplie at the average rate: P avg = = = (500 x 10 3 )(500A) = 50 MW The resistance of the transmission line: r = 0. /km So, = 0. /km x 1500 km = 330 Energy is issipate ue to at a rate: P avg = = 500 x 330 = 8.5 MW, about 33% f we increase the current to 65 A an reuce the voltage to 400 K, giving the same average rate of 50 MW: P avg = = 75 x 330 = 18.9 MW, about 5% Hence the general energy transmission rule: Transmit at the highest possible voltage an the lowest possible current.

33 Transformer: A evice can raise an lower the ac voltage in a circuit, keeping the prouct current voltage essentially constant. The ieal transformer consists of two coils, with ifferent numbers of turns, woun aroun an iron core. n the primary coil with N p : m sint The seconary coil with N s turns. The primary current magnetizing (very small) lags the primary voltage P by 90 0, so no power is elivere from the generator to the transformer (cos = 0). The small sinusoially changing primary current prouces a sinusoially changing magnetic flux B in the iron core, proucing an emf (B/t) in each turn of the seconary. This emf turn is the same in the primary an the seconary. Across the primary: p = turn N p ; Similarly, across n : s = turn N s. S P N N S P (transformation of f N s >N p, the evice is a step-up transformer: s > p. f N s <N p, it is a step-own transformer. A basic transformer circuit voltage)

34 So far, switch S is open, so no energy is transferre from the generator to the rest of the circuit. f we now close S to connect the seconary to the resistive loa : 1. An alternating current S appears, with energy issipation rate S = S /. S inuces an opposing emf in the primary winings 3. P of the primary cannot change in response to this opposing emf because it must always be equal to the emf provie by the generator 4. To maintain P, the generator prouces (in aition to mag ) an alternating current P in the primary circuit. The magnitue an phase constant of P are just those require for the emf inuce by P in the primary to exactly cancel the emf inuce there by S. Because the phase constant of P is not 90 0, so P can transfer energy (P avg = rms rms cos) to the primary.

35 Here eq is the value of the loa resistance as seen by the generator: S S P P f no energy is lost along the way, conservation of energy requires that P P S S P S P S P P S S N N N N N N 1 1 currents) (transformation of S P P S N N eq P P (equivalent resistance) N N S P eq

Solar Activity an Power-Gri Systems: A solar flare: A huge loop of e- an p extens

setting up a B fiel The transmission line, the groun, an the wires grouning the

An electrojet varies in both size an location, proucing an emf an a current.

36 Solar Activity an Power-Gri Systems: A solar flare: A huge loop of e- an p extens outwar from the Sun s surface The particles prouce a current, calle an electrojet, setting up a B fiel The transmission line, the groun, an the wires grouning the transformers form a conucting loop. An electrojet varies in both size an location, proucing an emf an a current. The current G (G: Geomagnetically nuce urrent) saturates the iron core, this isrupts the power transmission. Source: National Geographic

39 Homework: 1,, 7, 9, 10, 17, 3, 5, 9, 3, 38, 41, 48, 53 (pages )

PES 1120 Spring 2014, Spendier Lecture 36/Page 1

PES 1120 Spring 2014, Spendier Lecture 36/Page 1 PES 0 Spring 04, Spenier ecture 36/Page Toay: chapter 3 - R circuits: Dampe Oscillation - Driven series R circuit - HW 9 ue Wenesay - FQs Wenesay ast time you stuie the circuit (no resistance) The total